Modifying Heat Kernel Equation for Graphs
In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely:
$$ W_t(G) = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega_t(G) = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
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Just want a confirmation of my work :)
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In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely:
$$ W_t(G) = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega_t(G) = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
New contributor
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This question has an open bounty worth +50
reputation from Zachary Hunter ending in 5 days.
This question has not received enough attention.
Just want a confirmation of my work :)
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
– Zachary Hunter
Jan 4 at 20:08
How does the heat kernel equation in the title come into play in the question?
– mathreadler
2 days ago
1
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
– Zachary Hunter
2 days ago
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
– mathreadler
2 days ago
Heh, I was just waiting for him to go over to electrical flows, and he did.
– mathreadler
2 days ago
|
show 2 more comments
In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely:
$$ W_t(G) = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega_t(G) = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
New contributor
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
In spectral graph theory, I am aware that the following weight recurrence:
$$ w_t(v_i) = frac{1}{2}w_{t-1}(v_i)+sum_{v_j mid exists e_{ij} }frac{1}{2deg(v_i)} w_{t-1}(v_j) $$
Can be expressed in terms of eigen-vectors and eigenvalues of the Laplacian, $L=D-A$, nicely:
$$ W_t(G) = sum_{k=1}^n lambda_i^t a_i v_i $$
For the following recursion formula, would this equation work?
$$ omega_t(v_i) = sum_{v_j mid exists e_{ij} }frac{omega_{t-1}(v_j)}{deg(v_i)} $$
$$ Omega_t(G) = sum_{k=1}^n (2lambda_i-1)^t a_i v_i $$
My logic is that "$2lambda_i$" will double the weight, and the "$-1$" will subtract off the weight that a vertex directs back onto itself. This is not for a class, my school does not offer spectral graph theory.
spectral-graph-theory
spectral-graph-theory
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
New contributor
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
New contributor
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 days ago
Zachary Hunter
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
New contributor
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
asked Jan 4 at 5:13
Zachary HunterZachary Hunter
53110
53110
New contributor
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Zachary Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This question has an open bounty worth +50
reputation from Zachary Hunter ending in 5 days.
This question has not received enough attention.
Just want a confirmation of my work :)
This question has an open bounty worth +50
reputation from Zachary Hunter ending in 5 days.
This question has not received enough attention.
Just want a confirmation of my work :)
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
– Zachary Hunter
Jan 4 at 20:08
How does the heat kernel equation in the title come into play in the question?
– mathreadler
2 days ago
1
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
– Zachary Hunter
2 days ago
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
– mathreadler
2 days ago
Heh, I was just waiting for him to go over to electrical flows, and he did.
– mathreadler
2 days ago
|
show 2 more comments
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
– Zachary Hunter
Jan 4 at 20:08
How does the heat kernel equation in the title come into play in the question?
– mathreadler
2 days ago
1
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
– Zachary Hunter
2 days ago
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
– mathreadler
2 days ago
Heh, I was just waiting for him to go over to electrical flows, and he did.
– mathreadler
2 days ago
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
– Zachary Hunter
Jan 4 at 20:08
I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
– Zachary Hunter
Jan 4 at 20:08
How does the heat kernel equation in the title come into play in the question?
– mathreadler
2 days ago
How does the heat kernel equation in the title come into play in the question?
– mathreadler
2 days ago
1
1
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
– Zachary Hunter
2 days ago
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
– Zachary Hunter
2 days ago
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
– mathreadler
2 days ago
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
– mathreadler
2 days ago
Heh, I was just waiting for him to go over to electrical flows, and he did.
– mathreadler
2 days ago
Heh, I was just waiting for him to go over to electrical flows, and he did.
– mathreadler
2 days ago
|
show 2 more comments
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I got the first part from this video: “simons.berkeley.edu/events/openlectures2014-fall-4”
– Zachary Hunter
Jan 4 at 20:08
How does the heat kernel equation in the title come into play in the question?
– mathreadler
2 days ago
1
the first 5 minutes of the video I linked in the comments shows how this describes heat dispersion. if there's a more appropriate name, I'm all ears.
– Zachary Hunter
2 days ago
Ah ok, I did not see the link. Wow 79. That was like before C64 home computers. Must have been a big project making such computations back then.
– mathreadler
2 days ago
Heh, I was just waiting for him to go over to electrical flows, and he did.
– mathreadler
2 days ago